Lowest Landau level theory of the bosonic Jain states

TL;DR

This paper develops a microscopic non-commutative Landau level theory for bosonic Jain states in quantum Hall systems, capturing universal and non-universal properties without long-wavelength approximations.

Contribution

It introduces a novel approach to describe bosonic Jain states using non-commutative Landau levels and derives an effective non-commutative Chern-Simons theory.

Findings

Derived an approximate gap expression for bosonic Jain states.

Showed universal properties are encoded in a non-commutative Chern-Simons theory.

Connected the topological features to a non-commutative gauge theory.

Abstract

Quantum Hall systems offer the most familiar setting where strong inter-particle interactions combine with the topology of single particle states to yield novel phenomena. Despite our mature understanding of these systems, an open challenge has been to to develop a microscopic theory capturing both their universal and non-universal properties, when the Hamiltonian is restricted to the non-commutative space of the lowest Landau level. Here we develop such a theory for the Jain sequence of bosonic fractional quantum Hall states at fillings $\nu={p\over p+1}$. Building on a lowest Landau level description of a parent composite fermi liquid at $\nu = 1$, we describe how to dope the system to reach the Jain states. Upon doping, the composite fermions fill non-commutative generalizations of Landau levels, and the Jain states correspond to integer composite fermion filling. Using this approach, we obtain an approximate expression for the bosonic Jain sequence gaps with no reference to any long-wavelength approximation. Furthermore, we show that the universal properties, such as Hall conductivity, are encoded in an effective non-commutative Chern-Simons theory, which is obtained on integrating out the composite fermions. This theory has the same topological content as the familiar Abelian Chern-Simons theory on commutative space.